Problem: What is the slope of the line tangent to $f(x) = -2x^{2}+3x+7$ at $x = -1$ ?
Solution: The slope of the tangent line is $ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ $ = \lim_{h \to 0} \frac{(-2(x+h)^{2}+3(x+h)+7) - (-2x^{2}+3x+7)}{h}$ $ = \lim_{h \to 0} \frac{(-2(x^{2}+2x h+h^{2})+3(x+h)+7) - (-2x^{2}+3x+7)}{h}$ $ = \lim_{h \to 0} \frac{-2x^{2}-4(x h)-2h^{2}+3x+3h+7+2x^{2}-3x-7}{h}$ $ = \lim_{h \to 0} \frac{-4(x h)-2h^{2}+3h}{h}$ $ = \lim_{h \to 0} -4x-2h+3$ $ = -4x+3$ $ = (-4)(-1)+3$ $ = 7$